0.04/0.07	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.08	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.07/0.26	% Computer   : n003.cluster.edu
0.07/0.26	% Model      : x86_64 x86_64
0.07/0.26	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.07/0.26	% Memory     : 8042.1875MB
0.07/0.26	% OS         : Linux 3.10.0-693.el7.x86_64
0.07/0.26	% CPULimit   : 1200
0.07/0.26	% WCLimit    : 120
0.07/0.26	% DateTime   : Tue Jul 13 16:26:24 EDT 2021
0.07/0.26	% CPUTime    : 
101.58/13.05	% SZS status Theorem
101.58/13.05	
102.14/13.12	% SZS output start Proof
102.14/13.12	Take the following subset of the input axioms:
102.14/13.12	  fof(cn3, axiom, cn3 <=> ![P]: is_a_theorem(implies(implies(not(P), P), P))).
102.14/13.12	  fof(luka_cn3, conjecture, cn3).
102.14/13.12	  fof(luka_op_implies, axiom, op_implies).
102.14/13.12	  fof(luka_op_or, axiom, op_or).
102.14/13.12	  fof(modus_ponens, axiom, ![X, Y]: (is_a_theorem(Y) <= (is_a_theorem(implies(X, Y)) & is_a_theorem(X))) <=> modus_ponens).
102.14/13.12	  fof(op_and, axiom, op_and => ![X, Y]: not(or(not(X), not(Y)))=and(X, Y)).
102.14/13.12	  fof(op_equiv, axiom, op_equiv => ![X, Y]: and(implies(X, Y), implies(Y, X))=equiv(X, Y)).
102.14/13.12	  fof(op_implies_or, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y) <= op_implies_or).
102.14/13.12	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
102.14/13.12	  fof(or_2, axiom, or_2 <=> ![X, Y]: is_a_theorem(implies(Y, or(X, Y)))).
102.14/13.12	  fof(principia_modus_ponens, axiom, modus_ponens).
102.14/13.12	  fof(principia_op_and, axiom, op_and).
102.14/13.12	  fof(principia_op_equiv, axiom, op_equiv).
102.14/13.12	  fof(principia_op_implies_or, axiom, op_implies_or).
102.14/13.12	  fof(principia_r1, axiom, r1).
102.14/13.12	  fof(principia_r2, axiom, r2).
102.14/13.12	  fof(principia_r3, axiom, r3).
102.14/13.12	  fof(principia_r4, axiom, r4).
102.14/13.12	  fof(principia_r5, axiom, r5).
102.14/13.12	  fof(r1, axiom, ![P]: is_a_theorem(implies(or(P, P), P)) <=> r1).
102.14/13.12	  fof(r2, axiom, r2 <=> ![P, Q]: is_a_theorem(implies(Q, or(P, Q)))).
102.14/13.12	  fof(r3, axiom, r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
102.14/13.12	  fof(r4, axiom, r4 <=> ![P, Q, R]: is_a_theorem(implies(or(P, or(Q, R)), or(Q, or(P, R))))).
102.14/13.12	  fof(r5, axiom, ![P, Q, R]: is_a_theorem(implies(implies(Q, R), implies(or(P, Q), or(P, R)))) <=> r5).
102.14/13.12	  fof(substitution_of_equivalents, axiom, ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y) <=> substitution_of_equivalents).
102.14/13.12	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
102.14/13.12	
102.14/13.12	Now clausify the problem and encode Horn clauses using encoding 3 of
102.14/13.12	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
102.14/13.12	We repeatedly replace C & s=t => u=v by the two clauses:
102.14/13.12	  fresh(y, y, x1...xn) = u
102.14/13.12	  C => fresh(s, t, x1...xn) = v
102.14/13.12	where fresh is a fresh function symbol and x1..xn are the free
102.14/13.12	variables of u and v.
102.14/13.12	A predicate p(X) is encoded as p(X)=true (this is sound, because the
102.14/13.12	input problem has no model of domain size 1).
102.14/13.12	
102.14/13.12	The encoding turns the above axioms into the following unit equations and goals:
102.14/13.12	
102.14/13.12	Axiom 1 (luka_op_implies): op_implies = true.
102.14/13.12	Axiom 2 (principia_op_implies_or): op_implies_or = true.
102.14/13.12	Axiom 3 (luka_op_or): op_or = true.
102.14/13.12	Axiom 4 (principia_op_and): op_and = true.
102.14/13.12	Axiom 5 (principia_modus_ponens): modus_ponens = true.
102.14/13.12	Axiom 6 (principia_r1): r1 = true.
102.14/13.12	Axiom 7 (principia_r3): r3 = true.
102.14/13.12	Axiom 8 (principia_r5): r5 = true.
102.14/13.12	Axiom 9 (principia_r4): r4 = true.
102.14/13.12	Axiom 10 (principia_r2): r2 = true.
102.14/13.12	Axiom 11 (substitution_of_equivalents): substitution_of_equivalents = true.
102.14/13.12	Axiom 12 (principia_op_equiv): op_equiv = true.
102.14/13.12	Axiom 13 (cn3): fresh48(X, X) = true.
102.14/13.13	Axiom 14 (or_2): fresh17(X, X) = true.
102.14/13.13	Axiom 15 (modus_ponens_2): fresh60(X, X, Y) = true.
102.14/13.13	Axiom 16 (r1_1): fresh12(X, X, Y) = true.
102.14/13.13	Axiom 17 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
102.14/13.13	Axiom 18 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(is_a_theorem(Y), true, Z).
102.14/13.13	Axiom 19 (modus_ponens_2): fresh28(X, X, Y, Z) = is_a_theorem(Z).
102.14/13.13	Axiom 20 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
102.14/13.13	Axiom 21 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
102.14/13.13	Axiom 22 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
102.14/13.13	Axiom 23 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
102.14/13.13	Axiom 24 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
102.14/13.13	Axiom 25 (or_2_1): fresh16(X, X, Y, Z) = true.
102.14/13.13	Axiom 26 (r2_1): fresh10(X, X, Y, Z) = true.
102.14/13.13	Axiom 27 (r3_1): fresh8(X, X, Y, Z) = true.
102.14/13.13	Axiom 28 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
102.14/13.13	Axiom 29 (r4_1): fresh6(X, X, Y, Z, W) = true.
102.14/13.13	Axiom 30 (r5_1): fresh4(X, X, Y, Z, W) = true.
102.14/13.13	Axiom 31 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
102.14/13.13	Axiom 32 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
102.14/13.13	Axiom 33 (or_2_1): fresh16(or_2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
102.14/13.13	Axiom 34 (r2_1): fresh10(r2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
102.14/13.13	Axiom 35 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
102.14/13.13	Axiom 36 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
102.14/13.13	Axiom 37 (cn3_1): fresh47(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)).
102.14/13.13	Axiom 38 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
102.14/13.13	Axiom 39 (modus_ponens_2): fresh59(modus_ponens, true, X, Y) = fresh28(is_a_theorem(implies(X, Y)), true, X, Y).
102.14/13.13	Axiom 40 (or_2): fresh17(is_a_theorem(implies(y6, or(x6, y6))), true) = or_2.
102.14/13.13	Axiom 41 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
102.14/13.13	Axiom 42 (cn3): fresh48(is_a_theorem(implies(implies(not(p), p), p)), true) = cn3.
102.14/13.13	Axiom 43 (r5_1): fresh4(r5, true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
102.14/13.13	Axiom 44 (r4_1): fresh6(r4, true, X, Y, Z) = is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))).
102.14/13.13	
102.14/13.13	Lemma 45: fresh60(X, X, Y) = op_implies.
102.14/13.13	Proof:
102.14/13.13	  fresh60(X, X, Y)
102.14/13.13	= { by axiom 15 (modus_ponens_2) }
102.14/13.13	  true
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  op_implies
102.14/13.13	
102.14/13.13	Lemma 46: fresh28(is_a_theorem(implies(X, Y)), op_implies, X, Y) = fresh60(is_a_theorem(X), op_implies, Y).
102.14/13.13	Proof:
102.14/13.13	  fresh28(is_a_theorem(implies(X, Y)), op_implies, X, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh28(is_a_theorem(implies(X, Y)), true, X, Y)
102.14/13.13	= { by axiom 39 (modus_ponens_2) R->L }
102.14/13.13	  fresh59(modus_ponens, true, X, Y)
102.14/13.13	= { by axiom 5 (principia_modus_ponens) }
102.14/13.13	  fresh59(true, true, X, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh59(op_implies, true, X, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh59(op_implies, op_implies, X, Y)
102.14/13.13	= { by axiom 18 (modus_ponens_2) }
102.14/13.13	  fresh60(is_a_theorem(X), true, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh60(is_a_theorem(X), op_implies, Y)
102.14/13.13	
102.14/13.13	Lemma 47: or(not(X), Y) = implies(X, Y).
102.14/13.13	Proof:
102.14/13.13	  or(not(X), Y)
102.14/13.13	= { by axiom 23 (op_implies_or) R->L }
102.14/13.13	  fresh21(op_implies_or, true, X, Y)
102.14/13.13	= { by axiom 2 (principia_op_implies_or) }
102.14/13.13	  fresh21(true, true, X, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh21(op_implies, true, X, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh21(op_implies, op_implies, X, Y)
102.14/13.13	= { by axiom 22 (op_implies_or) }
102.14/13.13	  implies(X, Y)
102.14/13.13	
102.14/13.13	Lemma 48: not(implies(X, not(Y))) = and(X, Y).
102.14/13.13	Proof:
102.14/13.13	  not(implies(X, not(Y)))
102.14/13.13	= { by lemma 47 R->L }
102.14/13.13	  not(or(not(X), not(Y)))
102.14/13.13	= { by axiom 32 (op_and) R->L }
102.14/13.13	  fresh24(op_and, true, X, Y)
102.14/13.13	= { by axiom 4 (principia_op_and) }
102.14/13.13	  fresh24(true, true, X, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh24(op_implies, true, X, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh24(op_implies, op_implies, X, Y)
102.14/13.13	= { by axiom 20 (op_and) }
102.14/13.13	  and(X, Y)
102.14/13.13	
102.14/13.13	Lemma 49: is_a_theorem(implies(or(X, Y), or(Y, X))) = op_implies.
102.14/13.13	Proof:
102.14/13.13	  is_a_theorem(implies(or(X, Y), or(Y, X)))
102.14/13.13	= { by axiom 41 (r3_1) R->L }
102.14/13.13	  fresh8(r3, true, X, Y)
102.14/13.13	= { by axiom 7 (principia_r3) }
102.14/13.13	  fresh8(true, true, X, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh8(op_implies, true, X, Y)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh8(op_implies, op_implies, X, Y)
102.14/13.13	= { by axiom 27 (r3_1) }
102.14/13.13	  true
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  op_implies
102.14/13.13	
102.14/13.13	Lemma 50: is_a_theorem(implies(or(X, X), X)) = op_implies.
102.14/13.13	Proof:
102.14/13.13	  is_a_theorem(implies(or(X, X), X))
102.14/13.13	= { by axiom 35 (r1_1) R->L }
102.14/13.13	  fresh12(r1, true, X)
102.14/13.13	= { by axiom 6 (principia_r1) }
102.14/13.13	  fresh12(true, true, X)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh12(op_implies, true, X)
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  fresh12(op_implies, op_implies, X)
102.14/13.13	= { by axiom 16 (r1_1) }
102.14/13.13	  true
102.14/13.13	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.13	  op_implies
102.14/13.13	
102.14/13.13	Lemma 51: or(X, X) = X.
102.14/13.13	Proof:
102.14/13.13	  or(X, X)
102.14/13.13	= { by axiom 17 (substitution_of_equivalents_2) R->L }
102.14/13.13	  fresh(op_implies, op_implies, X, or(X, X))
102.14/13.13	= { by lemma 45 R->L }
102.14/13.13	  fresh(fresh60(op_implies, op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh(fresh60(true, op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 25 (or_2_1) R->L }
102.14/13.13	  fresh(fresh60(fresh16(op_implies, op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh(fresh60(fresh16(true, op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 14 (or_2) R->L }
102.14/13.13	  fresh(fresh60(fresh16(fresh17(op_implies, op_implies), op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh(fresh60(fresh16(fresh17(op_implies, true), op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh(fresh60(fresh16(fresh17(true, true), op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 26 (r2_1) R->L }
102.14/13.13	  fresh(fresh60(fresh16(fresh17(fresh10(op_implies, op_implies, x6, y6), true), op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh(fresh60(fresh16(fresh17(fresh10(op_implies, true, x6, y6), true), op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh(fresh60(fresh16(fresh17(fresh10(true, true, x6, y6), true), op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 10 (principia_r2) R->L }
102.14/13.13	  fresh(fresh60(fresh16(fresh17(fresh10(r2, true, x6, y6), true), op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 34 (r2_1) }
102.14/13.13	  fresh(fresh60(fresh16(fresh17(is_a_theorem(implies(y6, or(x6, y6))), true), op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 40 (or_2) }
102.14/13.13	  fresh(fresh60(fresh16(or_2, op_implies, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh(fresh60(fresh16(or_2, true, X, X), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 33 (or_2_1) }
102.14/13.13	  fresh(fresh60(is_a_theorem(implies(X, or(X, X))), op_implies, equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by lemma 46 R->L }
102.14/13.13	  fresh(fresh28(is_a_theorem(implies(implies(X, or(X, X)), equiv(X, or(X, X)))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 21 (op_equiv) R->L }
102.14/13.13	  fresh(fresh28(is_a_theorem(implies(implies(X, or(X, X)), fresh23(op_implies, op_implies, X, or(X, X)))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh(fresh28(is_a_theorem(implies(implies(X, or(X, X)), fresh23(op_implies, true, X, or(X, X)))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 1 (luka_op_implies) }
102.14/13.13	  fresh(fresh28(is_a_theorem(implies(implies(X, or(X, X)), fresh23(true, true, X, or(X, X)))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 12 (principia_op_equiv) R->L }
102.14/13.13	  fresh(fresh28(is_a_theorem(implies(implies(X, or(X, X)), fresh23(op_equiv, true, X, or(X, X)))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 36 (op_equiv) }
102.14/13.13	  fresh(fresh28(is_a_theorem(implies(implies(X, or(X, X)), and(implies(X, or(X, X)), implies(or(X, X), X)))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by lemma 48 R->L }
102.14/13.13	  fresh(fresh28(is_a_theorem(implies(implies(X, or(X, X)), not(implies(implies(X, or(X, X)), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by lemma 47 R->L }
102.14/13.13	  fresh(fresh28(is_a_theorem(implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by axiom 19 (modus_ponens_2) R->L }
102.14/13.13	  fresh(fresh28(fresh28(op_implies, op_implies, or(not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(X, or(X, X)))), implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by lemma 49 R->L }
102.14/13.13	  fresh(fresh28(fresh28(is_a_theorem(implies(or(not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(X, or(X, X)))), or(not(implies(X, or(X, X))), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))))))), op_implies, or(not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(X, or(X, X)))), implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.13	= { by lemma 47 }
102.14/13.13	  fresh(fresh28(fresh28(is_a_theorem(implies(or(not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(X, or(X, X)))), implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))))))), op_implies, or(not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(X, or(X, X)))), implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by lemma 46 }
102.14/13.15	  fresh(fresh28(fresh60(is_a_theorem(or(not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by lemma 47 }
102.14/13.15	  fresh(fresh28(fresh60(is_a_theorem(implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by axiom 19 (modus_ponens_2) R->L }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(op_implies, op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by lemma 45 R->L }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(fresh60(op_implies, op_implies, or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by lemma 49 R->L }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(fresh60(is_a_theorem(implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X)))))), op_implies, or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by lemma 46 R->L }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(fresh28(is_a_theorem(implies(implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))))), op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by lemma 47 R->L }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(fresh28(is_a_theorem(implies(implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), or(not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(X, or(X, X))))))), op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by lemma 47 R->L }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(fresh28(is_a_theorem(implies(or(not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), or(not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(X, or(X, X))))))), op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by axiom 44 (r4_1) R->L }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(fresh28(fresh6(r4, true, not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(or(X, X), X)), not(implies(X, or(X, X)))), op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by axiom 9 (principia_r4) }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(fresh28(fresh6(true, true, not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(or(X, X), X)), not(implies(X, or(X, X)))), op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(fresh28(fresh6(op_implies, true, not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(or(X, X), X)), not(implies(X, or(X, X)))), op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.15	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.15	  fresh(fresh28(fresh60(fresh28(fresh28(fresh6(op_implies, op_implies, not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), not(implies(or(X, X), X)), not(implies(X, or(X, X)))), op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by axiom 29 (r4_1) }
102.14/13.16	  fresh(fresh28(fresh60(fresh28(fresh28(true, op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.16	  fresh(fresh28(fresh60(fresh28(fresh28(op_implies, op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by axiom 19 (modus_ponens_2) }
102.14/13.16	  fresh(fresh28(fresh60(fresh28(is_a_theorem(or(not(implies(or(X, X), X)), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by lemma 47 }
102.14/13.16	  fresh(fresh28(fresh60(fresh28(is_a_theorem(implies(implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X)))))), op_implies, implies(or(X, X), X), implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by lemma 46 }
102.14/13.16	  fresh(fresh28(fresh60(fresh60(is_a_theorem(implies(or(X, X), X)), op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by lemma 50 }
102.14/13.16	  fresh(fresh28(fresh60(fresh60(op_implies, op_implies, implies(or(not(implies(X, or(X, X))), not(implies(or(X, X), X))), not(implies(X, or(X, X))))), op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by lemma 45 }
102.14/13.16	  fresh(fresh28(fresh60(op_implies, op_implies, implies(implies(X, or(X, X)), not(or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))))), op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by lemma 45 }
102.14/13.16	  fresh(fresh28(op_implies, op_implies, implies(X, or(X, X)), equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by axiom 19 (modus_ponens_2) }
102.14/13.16	  fresh(is_a_theorem(equiv(X, or(X, X))), op_implies, X, or(X, X))
102.14/13.16	= { by axiom 1 (luka_op_implies) }
102.14/13.16	  fresh(is_a_theorem(equiv(X, or(X, X))), true, X, or(X, X))
102.14/13.16	= { by axiom 38 (substitution_of_equivalents_2) R->L }
102.14/13.16	  fresh2(substitution_of_equivalents, true, X, or(X, X))
102.14/13.16	= { by axiom 11 (substitution_of_equivalents) }
102.14/13.16	  fresh2(true, true, X, or(X, X))
102.14/13.16	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.16	  fresh2(op_implies, true, X, or(X, X))
102.14/13.16	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.16	  fresh2(op_implies, op_implies, X, or(X, X))
102.14/13.16	= { by axiom 28 (substitution_of_equivalents_2) }
102.14/13.16	  X
102.14/13.16	
102.14/13.16	Lemma 52: implies(X, not(X)) = not(X).
102.14/13.16	Proof:
102.14/13.16	  implies(X, not(X))
102.14/13.16	= { by lemma 47 R->L }
102.14/13.16	  or(not(X), not(X))
102.14/13.16	= { by lemma 51 }
102.14/13.16	  not(X)
102.14/13.16	
102.14/13.16	Lemma 53: not(not(X)) = and(X, X).
102.14/13.16	Proof:
102.14/13.16	  not(not(X))
102.14/13.16	= { by lemma 52 R->L }
102.14/13.16	  not(implies(X, not(X)))
102.14/13.16	= { by lemma 48 }
102.14/13.16	  and(X, X)
102.14/13.16	
102.14/13.16	Lemma 54: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
102.14/13.16	Proof:
102.14/13.16	  implies(implies(X, not(Y)), Z)
102.14/13.16	= { by lemma 47 R->L }
102.14/13.16	  or(not(implies(X, not(Y))), Z)
102.14/13.16	= { by lemma 48 }
102.14/13.16	  or(and(X, Y), Z)
102.14/13.16	
102.14/13.16	Lemma 55: is_a_theorem(implies(implies(not(X), X), X)) = fresh47(cn3, op_implies, X).
102.14/13.16	Proof:
102.14/13.16	  is_a_theorem(implies(implies(not(X), X), X))
102.14/13.16	= { by axiom 37 (cn3_1) R->L }
102.14/13.16	  fresh47(cn3, true, X)
102.14/13.16	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.16	  fresh47(cn3, op_implies, X)
102.14/13.16	
102.14/13.16	Lemma 56: fresh60(is_a_theorem(implies(X, Y)), op_implies, implies(or(Z, X), or(Z, Y))) = is_a_theorem(implies(or(Z, X), or(Z, Y))).
102.14/13.16	Proof:
102.14/13.16	  fresh60(is_a_theorem(implies(X, Y)), op_implies, implies(or(Z, X), or(Z, Y)))
102.14/13.16	= { by lemma 46 R->L }
102.14/13.16	  fresh28(is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), op_implies, implies(X, Y), implies(or(Z, X), or(Z, Y)))
102.14/13.16	= { by axiom 43 (r5_1) R->L }
102.14/13.16	  fresh28(fresh4(r5, true, Z, X, Y), op_implies, implies(X, Y), implies(or(Z, X), or(Z, Y)))
102.14/13.16	= { by axiom 8 (principia_r5) }
102.14/13.16	  fresh28(fresh4(true, true, Z, X, Y), op_implies, implies(X, Y), implies(or(Z, X), or(Z, Y)))
102.14/13.16	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.16	  fresh28(fresh4(op_implies, true, Z, X, Y), op_implies, implies(X, Y), implies(or(Z, X), or(Z, Y)))
102.14/13.16	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.16	  fresh28(fresh4(op_implies, op_implies, Z, X, Y), op_implies, implies(X, Y), implies(or(Z, X), or(Z, Y)))
102.14/13.16	= { by axiom 30 (r5_1) }
102.14/13.16	  fresh28(true, op_implies, implies(X, Y), implies(or(Z, X), or(Z, Y)))
102.14/13.16	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.16	  fresh28(op_implies, op_implies, implies(X, Y), implies(or(Z, X), or(Z, Y)))
102.14/13.16	= { by axiom 19 (modus_ponens_2) }
102.14/13.16	  is_a_theorem(implies(or(Z, X), or(Z, Y)))
102.14/13.16	
102.14/13.16	Lemma 57: is_a_theorem(implies(or(X, Y), or(X, and(Y, Y)))) = op_implies.
102.14/13.16	Proof:
102.14/13.16	  is_a_theorem(implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by lemma 56 R->L }
102.14/13.16	  fresh60(is_a_theorem(implies(Y, and(Y, Y))), op_implies, implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by lemma 47 R->L }
102.14/13.16	  fresh60(is_a_theorem(or(not(Y), and(Y, Y))), op_implies, implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by axiom 19 (modus_ponens_2) R->L }
102.14/13.16	  fresh60(fresh28(op_implies, op_implies, or(and(Y, Y), not(Y)), or(not(Y), and(Y, Y))), op_implies, implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by lemma 49 R->L }
102.14/13.16	  fresh60(fresh28(is_a_theorem(implies(or(and(Y, Y), not(Y)), or(not(Y), and(Y, Y)))), op_implies, or(and(Y, Y), not(Y)), or(not(Y), and(Y, Y))), op_implies, implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by lemma 46 }
102.14/13.16	  fresh60(fresh60(is_a_theorem(or(and(Y, Y), not(Y))), op_implies, or(not(Y), and(Y, Y))), op_implies, implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by lemma 54 R->L }
102.14/13.16	  fresh60(fresh60(is_a_theorem(implies(implies(Y, not(Y)), not(Y))), op_implies, or(not(Y), and(Y, Y))), op_implies, implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by lemma 47 R->L }
102.14/13.16	  fresh60(fresh60(is_a_theorem(implies(or(not(Y), not(Y)), not(Y))), op_implies, or(not(Y), and(Y, Y))), op_implies, implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by lemma 50 }
102.14/13.16	  fresh60(fresh60(op_implies, op_implies, or(not(Y), and(Y, Y))), op_implies, implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by lemma 45 }
102.14/13.16	  fresh60(op_implies, op_implies, implies(or(X, Y), or(X, and(Y, Y))))
102.14/13.16	= { by lemma 45 }
102.14/13.16	  op_implies
102.14/13.16	
102.14/13.16	Goal 1 (luka_cn3): cn3 = true.
102.14/13.16	Proof:
102.14/13.16	  cn3
102.14/13.16	= { by axiom 42 (cn3) R->L }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), p)), true)
102.14/13.16	= { by lemma 55 }
102.14/13.16	  fresh48(fresh47(cn3, op_implies, p), true)
102.14/13.16	= { by axiom 1 (luka_op_implies) R->L }
102.14/13.16	  fresh48(fresh47(cn3, op_implies, p), op_implies)
102.14/13.16	= { by lemma 55 R->L }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), p)), op_implies)
102.14/13.16	= { by lemma 51 R->L }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), or(p, p))), op_implies)
102.14/13.16	= { by axiom 24 (op_or) R->L }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), fresh20(op_implies, op_implies, p, p))), op_implies)
102.14/13.16	= { by axiom 1 (luka_op_implies) }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), fresh20(op_implies, true, p, p))), op_implies)
102.14/13.16	= { by axiom 1 (luka_op_implies) }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), fresh20(true, true, p, p))), op_implies)
102.14/13.16	= { by axiom 3 (luka_op_or) R->L }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), fresh20(op_or, true, p, p))), op_implies)
102.14/13.16	= { by axiom 31 (op_or) }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), not(and(not(p), not(p))))), op_implies)
102.14/13.16	= { by lemma 53 R->L }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), not(not(not(not(p)))))), op_implies)
102.14/13.16	= { by lemma 53 }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), not(not(and(p, p))))), op_implies)
102.14/13.16	= { by lemma 53 }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(not(p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 52 R->L }
102.14/13.16	  fresh48(is_a_theorem(implies(implies(implies(p, not(p)), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 54 }
102.14/13.16	  fresh48(is_a_theorem(implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by axiom 19 (modus_ponens_2) R->L }
102.14/13.16	  fresh48(fresh28(op_implies, op_implies, implies(or(and(p, p), p), and(p, p)), implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 57 R->L }
102.14/13.16	  fresh48(fresh28(is_a_theorem(implies(or(not(or(and(p, p), p)), and(p, p)), or(not(or(and(p, p), p)), and(and(p, p), and(p, p))))), op_implies, implies(or(and(p, p), p), and(p, p)), implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 47 }
102.14/13.16	  fresh48(fresh28(is_a_theorem(implies(implies(or(and(p, p), p), and(p, p)), or(not(or(and(p, p), p)), and(and(p, p), and(p, p))))), op_implies, implies(or(and(p, p), p), and(p, p)), implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 47 }
102.14/13.16	  fresh48(fresh28(is_a_theorem(implies(implies(or(and(p, p), p), and(p, p)), implies(or(and(p, p), p), and(and(p, p), and(p, p))))), op_implies, implies(or(and(p, p), p), and(p, p)), implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 46 }
102.14/13.16	  fresh48(fresh60(is_a_theorem(implies(or(and(p, p), p), and(p, p))), op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by axiom 19 (modus_ponens_2) R->L }
102.14/13.16	  fresh48(fresh60(fresh28(op_implies, op_implies, implies(or(and(p, p), p), or(and(p, p), and(p, p))), implies(or(and(p, p), p), and(p, p))), op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 45 R->L }
102.14/13.16	  fresh48(fresh60(fresh28(fresh60(op_implies, op_implies, implies(or(not(or(and(p, p), p)), or(and(p, p), and(p, p))), or(not(or(and(p, p), p)), and(p, p)))), op_implies, implies(or(and(p, p), p), or(and(p, p), and(p, p))), implies(or(and(p, p), p), and(p, p))), op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 50 R->L }
102.14/13.16	  fresh48(fresh60(fresh28(fresh60(is_a_theorem(implies(or(and(p, p), and(p, p)), and(p, p))), op_implies, implies(or(not(or(and(p, p), p)), or(and(p, p), and(p, p))), or(not(or(and(p, p), p)), and(p, p)))), op_implies, implies(or(and(p, p), p), or(and(p, p), and(p, p))), implies(or(and(p, p), p), and(p, p))), op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 56 }
102.14/13.16	  fresh48(fresh60(fresh28(is_a_theorem(implies(or(not(or(and(p, p), p)), or(and(p, p), and(p, p))), or(not(or(and(p, p), p)), and(p, p)))), op_implies, implies(or(and(p, p), p), or(and(p, p), and(p, p))), implies(or(and(p, p), p), and(p, p))), op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.14/13.16	= { by lemma 47 }
102.14/13.16	  fresh48(fresh60(fresh28(is_a_theorem(implies(implies(or(and(p, p), p), or(and(p, p), and(p, p))), or(not(or(and(p, p), p)), and(p, p)))), op_implies, implies(or(and(p, p), p), or(and(p, p), and(p, p))), implies(or(and(p, p), p), and(p, p))), op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.60/13.17	= { by lemma 47 }
102.60/13.17	  fresh48(fresh60(fresh28(is_a_theorem(implies(implies(or(and(p, p), p), or(and(p, p), and(p, p))), implies(or(and(p, p), p), and(p, p)))), op_implies, implies(or(and(p, p), p), or(and(p, p), and(p, p))), implies(or(and(p, p), p), and(p, p))), op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.60/13.17	= { by lemma 46 }
102.60/13.17	  fresh48(fresh60(fresh60(is_a_theorem(implies(or(and(p, p), p), or(and(p, p), and(p, p)))), op_implies, implies(or(and(p, p), p), and(p, p))), op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.60/13.17	= { by lemma 57 }
102.60/13.17	  fresh48(fresh60(fresh60(op_implies, op_implies, implies(or(and(p, p), p), and(p, p))), op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.60/13.17	= { by lemma 45 }
102.60/13.17	  fresh48(fresh60(op_implies, op_implies, implies(or(and(p, p), p), and(and(p, p), and(p, p)))), op_implies)
102.60/13.17	= { by lemma 45 }
102.60/13.17	  fresh48(op_implies, op_implies)
102.60/13.17	= { by axiom 13 (cn3) }
102.60/13.17	  true
102.60/13.17	% SZS output end Proof
102.60/13.17	
102.60/13.17	RESULT: Theorem (the conjecture is true).
102.60/13.19	EOF